If \(G\)  is minimal, then by Lemma REF , the center of \(G\)  is compact.
For the other direction,
let \(\ltimes {u}{}\) , be a Levi decomposition of \(\) , with \(\)  a reductive \(k\) -group.
The central subgroup, \(Z()^\circ \)  is a \(k\) -torus [1]}, and we denote by \(\le Z()^\circ \)  the maximal \(k\) -split subtorus as in [1]}.
Set \(=\ltimes {u}{}\le \) .
We will show that \(A\)  is minimal and co-minimal in \(G\) .
Let \(\)  be a group topology on \(G\) , weaker than the analytic topology.
Let \(C=\lbrace t\in T : \forall u\in {u}{G},\ [t,u]=e\rbrace \) , since \(T\le Z(L)\) , \(C\)  is a central subgroup of \(G\)  and hence compact.
Moreover, by Proposition REF , the \(\) -subspace topology on \({u}{G}\)  is equal to the analytic topology. Hence by Corollary REF , the \(\) -subspace topology on \(A\)  is equal to the analytic topology, i.e. \(A\)  is relatively minimal in \(G\) .
In order to show co-minimality of \(A\) , we will show that the group \(G/A\)  is minimal. Observe that since \(\)  is a \(k\) -split solvable group then \((/)(k)\cong G/A\)  therefore, we may regard \(G/A\)  as \(k\) -points of the reductive \(k\) -algebraic group \(/\) . We claim that the center of \(L/T\cong G/A\)  is compact.
Let \(\pi \colon \rightarrow /\)  be the \(k\) -algebraic quotient map.
We claim that \(\pi (Z())=Z(/)\) , clearly \(\pi (Z())\subset Z(/)\) .
For the other direction, let \(s\in \)  be an element that is mapped into \(Z(/)\) , define the algebraic map
\(\mathbf {c}_s\colon \rightarrow , \ \ \mathbf {c}_s\colon l\mapsto [s,l]\) 
Observe that \(\mathbf {c}_s()\subset \mathcal {D}()\cap \)  and since \(\)  is connected, \(\mathbf {c}_s()\)  is connected.
Following [1]}, the intersection \( \mathcal {D}()\cap \)  is finite, hence by connectedness \(\mathbf {c}_s()=\lbrace e\rbrace \) , so \(s\in Z()\)  as needed.
By restricting the map \(\pi \)  to the central subgroup, we get a quotient map \(\pi \colon Z()\rightarrow Z(/)\) .
By [4]}, maximal \(k\) -split tori in \(Z(/)\)  are images of maximal \(k\) -split tori in \(Z()\) .
By the fact that \(\pi \)  maps \(\)  to \(\lbrace e\rbrace \) , we get that \(Z(/)\)  has no maximal \(k\) -split torus, implying that \(Z(L/T)\)  is compact.
Hence by Proposition REF , the group \(L/T\)  is minimal.
Finally, by applying Meson's Lemma REF  to \(A<G\) , since the \(\) -subspace topology on \(A\)  is analytic and \(G/A\cong L/T\)  is minimal, we get that the \(\)  topology on \(G\)  is equal to the analytic topology on \(G\) .
